Types of solving quadratic equations. Quadratic equations - examples with solutions, features and formulas

The discriminant, like quadratic equations, begins to be studied in an algebra course in the 8th grade. You can solve a quadratic equation through a discriminant and using Vieta's theorem. The method of studying quadratic equations, as well as discriminant formulas, is rather unsuccessfully taught to schoolchildren, like many things in real education. Therefore, school years pass, education in grades 9-11 is replaced by “higher education” and everyone is looking again - “How to solve a quadratic equation?”, “How to find the roots of the equation?”, “How to find the discriminant?” And...

Discriminant formula

The discriminant D of the quadratic equation a*x^2+bx+c=0 is equal to D=b^2–4*a*c.
The roots (solutions) of a quadratic equation depend on the sign of the discriminant (D):
D>0 – the equation has 2 different real roots;
D=0 - the equation has 1 root (2 matching roots):
D<0 – не имеет действительных корней (в школьной теории). В ВУЗах изучают комплексные числа и уже на множестве комплексных чисел уравнение с отрицательным дискриминантом имеет два комплексных корня.
The formula for calculating the discriminant is quite simple, so many websites offer an online discriminant calculator. We haven’t figured out this kind of scripts yet, so if anyone knows how to implement this, please write to us by email This email address is being protected from spambots. You must have JavaScript enabled to view it. .

General formula for finding the roots of a quadratic equation:

We find the roots of the equation using the formula
If the coefficient of a squared variable is paired, then it is advisable to calculate not the discriminant, but its fourth part
In such cases, the roots of the equation are found using the formula

The second way to find roots is Vieta's Theorem.

The theorem is formulated not only for quadratic equations, but also for polynomials. You can read this on Wikipedia or other electronic resources. However, to simplify, let’s consider the part that concerns the above quadratic equations, that is, equations of the form (a=1)
The essence of Vieta's formulas is that the sum of the roots of the equation is equal to the coefficient of the variable, taken with the opposite sign. The product of the roots of the equation is equal to the free term. Vieta's theorem can be written in formulas.
The derivation of Vieta's formula is quite simple. Let's write the quadratic equation through simple factors
As you can see, everything ingenious is simple at the same time. It is effective to use Vieta’s formula when the difference in modulus of the roots or the difference in the moduli of the roots is 1, 2. For example, the following equations, according to Vieta’s theorem, have roots




Up to equation 4, the analysis should look like this. The product of the roots of the equation is 6, therefore the roots can be the values ​​(1, 6) and (2, 3) or pairs with opposite signs. The sum of the roots is 7 (the coefficient of the variable with the opposite sign). From here we conclude that the solutions to the quadratic equation are x=2; x=3.
It is easier to select the roots of the equation among the divisors of the free term, adjusting their sign in order to fulfill the Vieta formulas. At first, this seems difficult to do, but with practice on a number of quadratic equations, this technique will turn out to be more effective than calculating the discriminant and finding the roots of the quadratic equation in the classical way.
As you can see, the school theory of studying the discriminant and methods of finding solutions to the equation is devoid of practical meaning - “Why do schoolchildren need a quadratic equation?”, “What is the physical meaning of the discriminant?”

Let's try to figure it out What does the discriminant describe?

In the algebra course they study functions, schemes for studying functions and constructing a graph of functions. Of all the functions, the parabola occupies an important place, the equation of which can be written in the form
So the physical meaning of the quadratic equation is the zeros of the parabola, that is, the points of intersection of the graph of the function with the abscissa axis Ox
I ask you to remember the properties of parabolas that are described below. The time will come to take exams, tests, or entrance exams and you will be grateful for the reference material. The sign of the squared variable corresponds to whether the branches of the parabola on the graph will go up (a>0),

or a parabola with branches down (a<0) .

The vertex of the parabola lies midway between the roots

Physical meaning of the discriminant:

If the discriminant is greater than zero (D>0) the parabola has two points of intersection with the Ox axis.
If the discriminant is zero (D=0) then the parabola at the vertex touches the x-axis.
And the last case, when the discriminant is less than zero (D<0) – график параболы принадлежит плоскости над осью абсцисс (ветки параболы вверх), или график полностью под осью абсцисс (ветки параболы опущены вниз).

Incomplete quadratic equations

Quadratic equations differ from linear equations in the presence of one unknown, raised to the second power. In the classical (canonical) form, the factors a, b and the free term c are not equal to zero.

A quadratic equation is an equation in which the left side is zero and the right side is a second-degree trinomial of the form:

Solving a trinomial or finding its roots means finding the values ​​of x at which the equality becomes true. It follows that the roots of such an equation are the values ​​of the variable x.

Finding roots using the discriminant formula

An example may have one or two roots, or it may have none. There is a very simple and understandable algorithm for determining the number of solutions. To do this, it is enough to find a discriminant - a special calculated value used when searching for roots. The formula for calculations is as follows:

Depending on the results obtained, the following conclusions can be drawn:

• there are two roots if D > 0;
• there is one solution if D = 0;
• there are no roots if D< 0.

If D ≥ 0, then you need to continue calculations using the formula:

The value of x1 will be equal to , and x2 - . If D = 0, then the sign “±” loses any meaning, because √0 = 0. In this case, the only root is equal to .

Examples of solving a quadratic equation

The algorithm for solving a polynomial is very simple:

1. Bring the expression to a classical form.
2. Determine whether there are roots of a quadratic equation (discriminant formula).
3. If D ≥ 0, then find the values ​​of the variable x using any of the known methods.

Let's give a clear example of how to solve a quadratic equation.

Problem 1. Find the roots and graphically indicate the solution area of ​​the equation 6x + 8 – 2×2 = 0.

First, it is necessary to bring the equality to the canonical form ax2+bx+c=0. To do this, we rearrange the terms of the polynomial.

Then, we simplify the expression by eliminating the coefficient in front of x2. Multiply the left and right sides by (-1)⁄2, the result is:

The advantages of formulas for finding the roots of a quadratic equation through a discriminant is that with their help you can solve any trinomial of the second degree.

So, in the given polynomial a=1, b=-3, and c=-4. Let's calculate the discriminant value for a specific example.

This means that the equation has two roots. To graphically find the solution area of ​​the example, you need to construct a parabola whose function is equal to .

The expression graphs will look like this:

In the example under consideration, D>0, therefore, there are two roots.

Tip 1: If the factor a is a negative number, you must multiply both sides of the example by (-1).

Tip 2: If there are fractions in the example, try to get rid of them by multiplying the left and right sides of the expression by their reciprocals.

Tip 3: You should always bring the equation to canonical form, this will help eliminate the possibility of confusion in the coefficients.

Vieta's theorem

There are methods that can significantly reduce the calculations. These include Vieta's theorem. This method cannot be applied to all types of equations, but only if the multiplier of the variable x2 is equal to one, that is, a = 1.

Let's look at this statement using specific examples:

1. 5×2 – 2x + 9 = 0 – application of the theorem in this case is inappropriate, since a = 5;
2. –x2 + 11x – 8 = 0 − a = -1, which means solving the equation using the Vieta method only after bringing it to the classical form, i.e., multiplying both sides by -1;
3. x2 + 4x – 5 = 0 – this task is ideal for analyzing the solution method.

In order to quickly find the roots of an expression, it is necessary to select a pair of x values ​​for which the following system of linear equations is valid.

Just. According to formulas and clear, simple rules. At the first stage

it is necessary to bring the given equation to a standard form, i.e. to the form:

If the equation is already given to you in this form, you do not need to do the first stage. The most important thing is to do it right

determine all the coefficients, A, b And c.

Formula for finding the roots of a quadratic equation.

The expression under the root sign is called discriminant . As you can see, to find X, we

we use only a, b and c. Those. coefficients from quadratic equation. Just carefully set it up

values a, b and c We calculate into this formula. We substitute with their signs!

For example, in the equation:

A =1; b = 3; c = -4.

We substitute the values ​​and write:

The example is almost solved:

This is the answer.

The most common mistakes are confusion with sign values a, b And With. Or rather, with substitution

negative values ​​into the formula for calculating the roots. A detailed recording of the formula comes to the rescue here

with specific numbers. If you have problems with calculations, do it!

Suppose we need to solve the following example:

Here a = -6; b = -5; c = -1

We describe everything in detail, carefully, without missing anything with all the signs and brackets:

Quadratic equations often look slightly different. For example, like this:

Now take note of practical techniques that dramatically reduce the number of errors.

First appointment. Don't be lazy before solving a quadratic equation bring it to standard form.

What does this mean?

Let's say that after all the transformations you get the following equation:

Don't rush to write the root formula! You'll almost certainly get the odds mixed up a, b and c.

Construct the example correctly. First, X squared, then without square, then the free term. Like this:

Get rid of the minus. How? We need to multiply the entire equation by -1. We get:

But now you can safely write down the formula for the roots, calculate the discriminant and finish solving the example.

Decide for yourself. You should now have roots 2 and -1.

Reception second. Check the roots! By Vieta's theorem.

To solve the given quadratic equations, i.e. if the coefficient

x 2 +bx+c=0,

Thenx 1 x 2 =c

x 1 +x 2 =−b

For a complete quadratic equation in which a≠1:

x 2 +bx+c=0,

divide the whole equation by A:

→ →

Where x 1 And x 2 - roots of the equation.

Reception third. If your equation has fractional coefficients, get rid of the fractions! Multiply

equation with a common denominator.

Conclusion. Practical tips:

1. Before solving, we bring the quadratic equation to standard form and build it Right.

2. If there is a negative coefficient in front of the X squared, we eliminate it by multiplying everything

equations by -1.

3. If the coefficients are fractional, we eliminate the fractions by multiplying the entire equation by the corresponding

factor.

4. If x squared is pure, its coefficient is equal to one, the solution can be easily checked by

Quadratic equation - easy to solve! *Hereinafter referred to as “KU”. Friends, it would seem that there could be nothing simpler in mathematics than solving such an equation. But something told me that many people have problems with him. I decided to see how many on-demand impressions Yandex gives out per month. Here's what happened, look:

What does it mean? This means that about 70,000 people a month are looking for this information, and this is summer, and what will happen during the school year - there will be twice as many requests. This is not surprising, because those guys and girls who graduated from school a long time ago and are preparing for the Unified State Exam are looking for this information, and schoolchildren also strive to refresh their memory.

Despite the fact that there are a lot of sites that tell you how to solve this equation, I decided to also contribute and publish the material. Firstly, I want visitors to come to my site based on this request; secondly, in other articles, when the topic of “KU” comes up, I will provide a link to this article; thirdly, I’ll tell you a little more about his solution than is usually stated on other sites. Let's get started! The content of the article:

A quadratic equation is an equation of the form:

where coefficients a,band c are arbitrary numbers, with a≠0.

In the school course, the material is given in the following form - the equations are divided into three classes:

1. They have two roots.

2. *Have only one root.

3. They have no roots. It is worth especially noting here that they do not have real roots

How are roots calculated? Just!

We calculate the discriminant. Underneath this “terrible” word lies a very simple formula:

The root formulas are as follows:

*You need to know these formulas by heart.

You can immediately write down and solve:

Example:

1. If D > 0, then the equation has two roots.

2. If D = 0, then the equation has one root.

3. If D< 0, то уравнение не имеет действительных корней.

Let's look at the equation:

In this regard, when the discriminant is equal to zero, the school course says that one root is obtained, here it is equal to nine. Everything is correct, it is so, but...

This idea is somewhat incorrect. In fact, there are two roots. Yes, yes, don’t be surprised, you get two equal roots, and to be mathematically precise, then the answer should write two roots:

x 1 = 3 x 2 = 3

But this is so - a small digression. At school you can write it down and say that there is one root.

Now the next example:

As we know, the root of a negative number cannot be taken, so there is no solution in this case.

That's the whole decision process.

Quadratic function.

This shows what the solution looks like geometrically. This is extremely important to understand (in the future, in one of the articles we will analyze in detail the solution to the quadratic inequality).

This is a function of the form:

where x and y are variables

a, b, c – given numbers, with a ≠ 0

The graph is a parabola:

That is, it turns out that by solving a quadratic equation with “y” equal to zero, we find the points of intersection of the parabola with the x axis. There can be two of these points (the discriminant is positive), one (the discriminant is zero) and none (the discriminant is negative). Details about the quadratic function You can view article by Inna Feldman.

Let's look at examples:

Example 1: Solve 2x 2 +8 x–192=0

a=2 b=8 c= –192

D=b 2 –4ac = 8 2 –4∙2∙(–192) = 64+1536 = 1600

Answer: x 1 = 8 x 2 = –12

*It was possible to immediately divide the left and right sides of the equation by 2, that is, simplify it. The calculations will be easier.

Example 2: Decide x 2–22 x+121 = 0

a=1 b=–22 c=121

D = b 2 –4ac =(–22) 2 –4∙1∙121 = 484–484 = 0

We found that x 1 = 11 and x 2 = 11

It is permissible to write x = 11 in the answer.

Answer: x = 11

Example 3: Decide x 2 –8x+72 = 0

a=1 b= –8 c=72

D = b 2 –4ac =(–8) 2 –4∙1∙72 = 64–288 = –224

The discriminant is negative, there is no solution in real numbers.

Answer: no solution

The discriminant is negative. There is a solution!

Here we will talk about solving the equation in the case when a negative discriminant is obtained. Do you know anything about complex numbers? I will not go into detail here about why and where they arose and what their specific role and necessity in mathematics is; this is a topic for a large separate article.

The concept of a complex number.

A little theory.

A complex number z is a number of the form

z = a + bi

where a and b are real numbers, i is the so-called imaginary unit.

a+bi – this is a SINGLE NUMBER, not an addition.

The imaginary unit is equal to the root of minus one:

Now consider the equation:

We get two conjugate roots.

Incomplete quadratic equation.

Let's consider special cases, this is when the coefficient “b” or “c” is equal to zero (or both are equal to zero). They can be solved easily without any discriminants.

Case 1. Coefficient b = 0.

The equation becomes:

Let's transform:

Example:

4x 2 –16 = 0 => 4x 2 =16 => x 2 = 4 => x 1 = 2 x 2 = –2

Case 2. Coefficient c = 0.

The equation becomes:

Let's transform and factorize:

*The product is equal to zero when at least one of the factors is equal to zero.

Example:

9x 2 –45x = 0 => 9x (x–5) =0 => x = 0 or x–5 =0

x 1 = 0 x 2 = 5

Case 3. Coefficients b = 0 and c = 0.

Here it is clear that the solution to the equation will always be x = 0.

Useful properties and patterns of coefficients.

There are properties that allow you to solve equations with large coefficients.

Ax 2 + bx+ c=0 equality holds

a + b+ c = 0, That

- if for the coefficients of the equation Ax 2 + bx+ c=0 equality holds

a+ c =b, That

These properties help solve a certain type of equation.

Example 1: 5001 x 2 –4995 x – 6=0

The sum of the odds is 5001+( – 4995)+(– 6) = 0, which means

Example 2: 2501 x 2 +2507 x+6=0

Equality holds a+ c =b, Means

Regularities of coefficients.

1. If in the equation ax 2 + bx + c = 0 the coefficient “b” is equal to (a 2 +1), and the coefficient “c” is numerically equal to the coefficient “a”, then its roots are equal

ax 2 + (a 2 +1)∙x+ a= 0 = > x 1 = –a x 2 = –1/a.

Example. Consider the equation 6x 2 + 37x + 6 = 0.

x 1 = –6 x 2 = –1/6.

2. If in the equation ax 2 – bx + c = 0 the coefficient “b” is equal to (a 2 +1), and the coefficient “c” is numerically equal to the coefficient “a”, then its roots are equal

ax 2 – (a 2 +1)∙x+ a= 0 = > x 1 = a x 2 = 1/a.

Example. Consider the equation 15x 2 –226x +15 = 0.

x 1 = 15 x 2 = 1/15.

3. If in Eq. ax 2 + bx – c = 0 coefficient “b” is equal to (a 2 – 1), and coefficient “c” is numerically equal to the coefficient “a”, then its roots are equal

ax 2 + (a 2 –1)∙x – a= 0 = > x 1 = – a x 2 = 1/a.

Example. Consider the equation 17x 2 +288x – 17 = 0.

x 1 = – 17 x 2 = 1/17.

4. If in the equation ax 2 – bx – c = 0 the coefficient “b” is equal to (a 2 – 1), and the coefficient c is numerically equal to the coefficient “a”, then its roots are equal

ax 2 – (a 2 –1)∙x – a= 0 = > x 1 = a x 2 = – 1/a.

Example. Consider the equation 10x 2 – 99x –10 = 0.

x 1 = 10 x 2 = – 1/10

Vieta's theorem.

Vieta's theorem is named after the famous French mathematician Francois Vieta. Using Vieta's theorem, we can express the sum and product of the roots of an arbitrary KU in terms of its coefficients.

45 = 1∙45 45 = 3∙15 45 = 5∙9.

In total, the number 14 gives only 5 and 9. These are the roots. With a certain skill, using the presented theorem, you can solve many quadratic equations orally immediately.

Vieta's theorem, in addition. It is convenient in that after solving a quadratic equation in the usual way (through a discriminant), the resulting roots can be checked. I recommend doing this always.

TRANSPORTATION METHOD

With this method, the coefficient “a” is multiplied by the free term, as if “thrown” to it, which is why it is called "transfer" method. This method is used when the roots of the equation can be easily found using Vieta's theorem and, most importantly, when the discriminant is an exact square.

If A± b+c≠ 0, then the transfer technique is used, for example:

2X 2 – 11x+ 5 = 0 (1) => X 2 – 11x+ 10 = 0 (2)

Using Vieta's theorem in equation (2), it is easy to determine that x 1 = 10 x 2 = 1

The resulting roots of the equation must be divided by 2 (since the two were “thrown” from x 2), we get

x 1 = 5 x 2 = 0.5.

What is the rationale? Look what's happening.

The discriminants of equations (1) and (2) are equal:

If you look at the roots of the equations, you only get different denominators, and the result depends precisely on the coefficient of x 2:

The second (modified) one has roots that are 2 times larger.

Therefore, we divide the result by 2.

*If we reroll the three, we will divide the result by 3, etc.

Answer: x 1 = 5 x 2 = 0.5

Sq. ur-ie and Unified State Examination.

I’ll tell you briefly about its importance - YOU MUST BE ABLE TO DECIDE quickly and without thinking, you need to know the formulas of roots and discriminants by heart. Many of the problems included in the Unified State Examination tasks boil down to solving a quadratic equation (geometric ones included).

Something worth noting!

1. The form of writing an equation can be “implicit”. For example, the following entry is possible:

15+ 9x 2 - 45x = 0 or 15x+42+9x 2 - 45x=0 or 15 -5x+10x 2 = 0.

You need to bring it to a standard form (so as not to get confused when solving).

2. Remember that x is an unknown quantity and it can be denoted by any other letter - t, q, p, h and others.

", that is, equations of the first degree. In this lesson we will look at what is called a quadratic equation and how to solve it.

What is a quadratic equation?

Important!

The degree of an equation is determined by the highest degree to which the unknown stands.

If the maximum power in which the unknown is “2”, then you have a quadratic equation.

Examples of quadratic equations

• 5x 2 − 14x + 17 = 0
• −x 2 + x +

  1

  3

  = 0
• x 2 + 0.25x = 0
• x 2 − 8 = 0

Important! The general form of a quadratic equation looks like this:

A x 2 + b x + c = 0

• “a” is the first or highest coefficient;
• “b” is the second coefficient;
• “c” is a free term.

To find “a”, “b” and “c” you need to compare your equation with the general form of the quadratic equation “ax 2 + bx + c = 0”.

Let's practice determining the coefficients "a", "b" and "c" in quadratic equations.

• a = −1
• b = 1
• c =

  1

  3

• a = 1
• b = 0.25
• c = 0

• a = 1
• b = 0
• c = −8

How to Solve Quadratic Equations

Unlike linear equations, a special method is used to solve quadratic equations. formula for finding roots.

Remember!

To solve a quadratic equation you need:

• bring the quadratic equation to the general form “ax 2 + bx + c = 0”.
• That is, only “0” should remain on the right side;

use formula for roots:

Let's look at an example of how to use the formula to find the roots of a quadratic equation. Let's solve a quadratic equation.

X 2 − 3x − 4 = 0 The equation “x 2 − 3x − 4 = 0” has already been reduced to the general form “ax 2 + bx + c = 0” and does not require additional simplifications. To solve it, we just need to apply.

formula for finding the roots of a quadratic equation

x 1;2 =

It can be used to solve any quadratic equation.
In the formula “x 1;2 = ” the radical expression is often replaced

“b 2 − 4ac” for the letter “D” and is called discriminant. The concept of a discriminant is discussed in more detail in the lesson “What is a discriminant”.

Let's look at another example of a quadratic equation.

x 2 + 9 + x = 7x

In this form, it is quite difficult to determine the coefficients “a”, “b” and “c”. Let's first reduce the equation to the general form “ax 2 + bx + c = 0”.
X 2 + 9 + x = 7x
x 2 + 9 + x − 7x = 0
x 2 + 9 − 6x = 0

x 2 − 6x + 9 = 0

Now you can use the formula for the roots.
X 1;2 =
X 1;2 =
X 1;2 =
x 1;2 =

There are times when quadratic equations have no roots. This situation occurs when the formula contains a negative number under the root.